Network Science is the emerging field concerned with the study of large, realistic networks. This interdisciplinary endeavor, focusing on the patterns of interactions that arise between individual components of natural and engineered systems, has been applied to data sets from activities as diverse as high-throughput biological experiments, online trading information, smart-meter utility supplies, and pervasive telecommunications and surveillance technologies. This unique text/reference provides a fascinating insight into the state of the art in network science, highlighting the commonality across very different areas of application and the ways in which each area can be advanced by injecting ideas and techniques from another. The book includes contributions from an international selection of experts, providing viewpoints from a broad range of disciplines. It emphasizes networks that arise in nature-such as food webs, protein interactions, gene expression, and neural connections-and in technology-such as finance, airline transport, urban development and global trade. Topics and Features: begins with a clear overview chapter to introduce this interdisciplinary field; discusses the classic network science of fixed connectivity structures, including empirical studies, mathematical models and computational algorithms; examines time-dependent processes that take place over networks, covering topics such as synchronisation, and message passing algorithms; investigates time-evolving networks, such as the World Wide Web and shifts in topological properties (connectivity, spectrum, percolation); explores applications of complex networks in the physical and engineering sciences, looking ahead to new developments in the field. Researchers and professionals from disciplines as varied as computer science, mathematics, engineering, physics, chemistry, biology, ecology, neuroscience, epidemiology, and the social sciences will all benefit from this topical and broad overview of current activities and grand challenges in the unfolding field of network science.

Contents and treatment are fresh and very different from the standard treatments Presents a fully constructive version of what it means to do algebra The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader

Asymptotic analysis of stochastic stock price models is the central topic of the present volume. Special examples of such models are stochastic volatility models, that have been developed as an answer to certain imperfections in a celebrated Black-Scholes model of option pricing. In a stock price model with stochastic volatility, the random behavior of the volatility is described by a stochastic process. For instance, in the Hull-White model the volatility process is a geometric Brownian motion, the Stein-Stein model uses an Ornstein-Uhlenbeck process as the stochastic volatility, and in the Heston model a Cox-Ingersoll-Ross process governs the behavior of the volatility. One of the author's main goals is to provide sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions, and implied volatilities in various stochastic volatility models. The author also establishes sharp asymptotic formulas for the implied volatility at extreme strikes in general stochastic stock price models. The present volume is addressed to researchers and graduate students working in the area of financial mathematics, analysis, or probability theory. The reader is expected to be familiar with elements of classical analysis, stochastic analysis and probability theory.

This proceedings volume is based on papers presented at the First Annual Workshop on Inverse Problems which was held in June 2011 at the Department of Mathematics, Chalmers University of Technology. The purpose of the workshop was to present new analytical developments and numerical methods for solutions of inverse problems. State-of-the-art and future challenges in solving inverse problems for a broad range of applications was also discussed. The contributions in this volume are reflective of these themes and will be beneficial to researchers in this area.

Based on a two-semester course aimed at illustrating various interactions of "pure mathematics" with other sciences, such as hydrodynamics, thermodynamics, statistical physics and information theory, this text unifies three general topics of analysis and physics, which are as follows: the dimensional analysis of physical quantities, which contains various applications including Kolmogorov's model for turbulence; functions of very large number of variables and the principle of concentration along with the non-linear law of large numbers, the geometric meaning of the Gauss and Maxwell distributions, and the Kotelnikov-Shannon theorem; and, finally, classical thermodynamics and contact geometry, which covers two main principles of thermodynamics in the language of differential forms, contact distributions, the Frobenius theorem and the Carnot-Caratheodory metric. It includes problems, historical remarks, and Zorich's popular article, "Mathematics as language and method."

This selection of outstanding articles - an outgrowth of the QMath9 meeting for young scientists - covers new techniques and recent results on spectral theory, statistical mechanics, Bose-Einstein condensation, random operators, magnetic Schroedinger operators and more. The book's pedagogical style makes it a useful introduction to the research literature for postgraduate students. For more expert researchers it will serve as a concise source of modern reference.

Matrix-valued data sets - so-called second order tensor fields - have gained significant importance in scientific visualization and image processing due to recent developments such as diffusion tensor imaging. This book is the first edited volume that presents the state of the art in the visualization and processing of tensor fields. It contains some longer chapters dedicated to surveys and tutorials of specific topics, as well as a great deal of original work by leading experts that has not been published before. It serves as an overview for the inquiring scientist, as a basic foundation for developers and practitioners, and as as a textbook for specialized classes and seminars for graduate and doctoral students.

Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook not only follows this programme, but additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

A long long time ago, echoing philosophical and aesthetic principles that existed since antiquity, William of Ockham enounced the principle of parsimony, better known today as Ockham's razor: "Entities should not be multiplied without neces sity. " This principle enabled scientists to select the "best" physical laws and theories to explain the workings of the Universe and continued to guide scienti?c research, leadingtobeautifulresultsliketheminimaldescriptionlength approachtostatistical inference and the related Kolmogorov complexity approach to pattern recognition. However, notions of complexity and description length are subjective concepts anddependonthelanguage"spoken"whenpresentingideasandresults. The?eldof sparse representations, that recently underwent a Big Bang like expansion, explic itly deals with the Yin Yang interplay between the parsimony of descriptions and the "language" or "dictionary" used in them, and it became an extremely exciting area of investigation. It already yielded a rich crop of mathematically pleasing, deep and beautiful results that quickly translated into a wealth of practical engineering applications. You are holding in your hands the ?rst guide book to Sparseland, and I am sure you'll ?nd in it both familiar and new landscapes to see and admire, as well as ex cellent pointers that will help you ?nd further valuable treasures. Enjoy the journey to Sparseland! Haifa, Israel, December 2009 Alfred M. Bruckstein vii Preface This book was originally written to serve as the material for an advanced one semester (fourteen 2 hour lectures) graduate course for engineering students at the Technion, Israel.

This International Conference on Clifford AlgebrfU and Their Application, in Math ematical Phy,ic, is the third in a series of conferences on this theme, which started at the Univer,ity of Kent in Canterbury in 1985 and was continued at the Univer,iU de, Science, et Technique, du Languedoc in Montpellier in 1989. Since the start of this series of Conferences the research fields under consideration have evolved quite a lot. The number of scientific papers on Clifford Algebra, Clifford Analysis and their impact on the modelling of physics phenomena have increased tremendously and several new books on these topics were published. We were very pleased to see old friends back and to wellcome new guests who by their inspiring talks contributed fundamentally to tracing new paths for the future development of this research area. The Conference was organized in Deinze, a small rural town in the vicinity of the University town Gent. It was hosted by De Ceder, a vacation and seminar center in a green area, a typical landscape of Flanders's "plat pays" . The Conference was attended by 61 participants coming from 18 countries; there were 10 main talks on invitation, 37 contributions accepted by the Organizing Com mittee and a poster session. There was also a book display of Kluwer Academic Publishers. As in the Proceedings of the Canterbury and Montpellier conferences we have grouped the papers accordingly to the themes they are related to: Clifford Algebra, Clifford Analysis, Classical Mechanics, Mathematical Physics and Physics Models.

In these proceedings basic questions regarding n-body Schr|dinger operators are dealt with, such as asymptotic completeness of systems with long-range potentials (including Coulomb), a new proof of completeness for short-range potentials, energy asymptotics of large Coulomb systems,asymptotic neutrality of polyatomic molecules. Other contributions deal withdifferent types of problems, such as quantum stability, Schr|dinger operators on a torus and KAM theory, semiclassical theory, time delay, radiation conditions, magnetic Stark resonances, random Schr|dinger operators and stochastic spectral analysis. The volume presents the results in such detail that it could well serve as basic literature for seminar work.

The shared purpose in this collection of papers is to apply the theory of self-adjoint extensions of symmetry operators in various areas of physics. This allows the construction of exactly solvable models in quantum mechanics, quantum field theory, high energy physics, solid-state physics, microelectronics and other fields. The 20 papers selected for these proceedings give an overview of this field of research unparallelled in the published literature; in particular the views of the leading schools are clearly presented. The book will be an important source for researchers and graduate students in mathematical physics for many years to come. In these proceedings, researchers and graduate students in mathematical physics will find ways to construct exactly solvable models in quantum mechanics, quantum field theory, high energy physics, solid-state physics, microelectronics and other fields.

Mainly drawing on explicit examples, the author introduces the reader to themost recent techniques to study finite and infinite dynamical systems. Without any knowledge of differential geometry or lie groups theory the student can follow in a series of case studies the most recent developments. r-matrices for Calogero-Moser systems and Toda lattices are derived. Lax pairs for nontrivial infinite dimensionalsystems are constructed as limits of classical matrix algebras. The reader will find explanations of the approach to integrable field theories, to spectral transform methods and to solitons. New methods are proposed, thus helping students not only to understand established techniques but also to interest them in modern research on dynamical systems.

Dirac's formalism of quantum mechanics was always praised for its elegance. This book introduces the student to its mathematical foundations and demonstrates its ease of applicability to problems in quantum physics. The book starts by describing in detail the concept of Gel'fand triplets and how one can make use of them to make the Dirac heuristic approach rigorous. The results are then deepened by giving the analytic tools, such as the Hardy class function and Hilbert and Mellin transforms, needed in applications to physical problems. Next, the RHS model for decaying states based on the concept of Gamow vectors is presented. Applications are given to physical theories of such phenomena as decaying states and resonances.

This book contains survey papers based on the lectures presented at the 3rd International Winter School "Modern Problems of Mathematics and Mechanics" held in January 2010 at the Belarusian State University, Minsk. These lectures are devoted to different problems of modern analysis and its applications. An extended presentation of modern problems of applied analysis will enable the reader to get familiar with new approaches of mostly interdisciplinary character. The results discussed are application oriented and present new insight into applied problems of growing importance such as applications to composite materials, anomalous diffusion, and fluid dynamics.

For experiments, dimensional analysis enables the design, checks the validity, orders the procedure and synthesises the data. Additionally it can provide relationships between variables where standard analysis is not available. This widely valuable analysis for engineers and scientists is here presented to the student, the teacher and the researcher. It is the first complete modern text that covers developments over the last three decades while closing all outstanding logical gaps. Dimensional Analysis also lists the logical stages of the analysis, so showing clearly the care to be taken in its use while revealing the very few limitations of application. As the conclusion of that logic, it gives the author's original proof of the fundamental and only theorem. Unlike past texts, Dimensional Analysis includes examples for which the answer does not already exist from standard analysis. It also corrects the many errors present in the existing literature by including accurate solutions. Dimensional Analysis is written for all branches of engineering and science as a teaching book covering both undergraduate and postgraduate courses, as a guide for the lecturer and as a reference volume for the researcher.

Asymptotic Geometric Analysis is concerned with the geometric and linear properties of finite dimensional objects, normed spaces, and convex bodies, especially with the asymptotics of their various quantitative parameters as the dimension tends to infinity. The deep geometric, probabilistic, and combinatorial methods developed here are used outside the field in many areas of mathematics and mathematical sciences. The Fields Institute Thematic Program in the Fall of 2010 continued an established tradition of previous large-scale programs devoted to the same general research direction. The main directions of the program included: * Asymptotic theory of convexity and normed spaces * Concentration of measure and isoperimetric inequalities, optimal transportation approach * Applications of the concept of concentration * Connections with transformation groups and Ramsey theory * Geometrization of probability * Random matrices * Connection with asymptotic combinatorics and complexity theory These directions are represented in this volume and reflect the present state of this important area of research. It will be of benefit to researchers working in a wide range of mathematical sciences-in particular functional analysis, combinatorics, convex geometry, dynamical systems, operator algebras, and computer science.

Dieses Buch nimmt Sie mit auf eine spannende Reise durch die Welt der Wissenschaft: von den Fallgesetzen des Galilei bis zu Einsteins Gravitationswellen, von Newtons Axiomen bis zum Wasserstoffatom, von der naturlichen Auslese bis zum Schwarmverhalten, von der Skala der Empfindungen bis zu den Grenzen des Wachstums auf unserem Planeten. Sie lernen Differentialgleichungen als machtiges Instrument kennen, das die Mathematik zur Erforschung der Natur bereitstellt. Ihre Loesungen enthullen, um mit Laplace zu sprechen, die Bewegungen der groessten Weltkoerper und des kleinsten Atoms und vieles von dem, was dazwischen liegt - einschliesslich unserer selbst. Lassen Sie sich von Wolfgang Tschirk begeistern: Er gewahrt Ihnen einen unterhaltsamen Blick auf die Verstandesleistungen jener, die dem Laplaceschen Damon Stuck fur Stuck sein Geheimnis ablauschen. Um Macht und Schoenheit der Differentialgleichungen zu erleben, sollten Sie Affinitat zur Mathematik mitbringen und auch vor Formeln nicht zuruckschrecken. Aber Sie werden sehen: Entgegen ihrem Ruf sind Differentialgleichungen im Grunde leicht zu verstehen und oft sogar leicht zu loesen.

The techniques presented here are useful for solving mathematical contest problems in algebra and analysis. Most of the examples and exercises that appear in the book originate from mathematical Olympiad competitions around the world. In the first four chapters the authors cover material for competitions at high school level. The level advances with the chapters. The topics explored include polynomials, functional equations, sequences and an elementary treatment of complex numbers. The final chapters provide a comprehensive list of problems posed at national and international contests in recent years, and solutions to all exercises and problems presented in the book. It helps students in preparing for national and international mathematical contests form high school level to more advanced competitions and will also be useful for their first year of mathematical studies at the university. It will be of interest to teachers in college and university level, and trainers of the mathematical Olympiads.

This text provides an introduction to the topic of transcendental numbers for undergraduate and graduate students. Chapters 1 to 18 can be used for an introductory course in transcendental numbers aimed at senior undergraduates and first-year graduate students. Since the Schneider-Lang theorem is the main focus, the reader must have a rudimentary background in complex analysis. Some of the essential features of elliptic curve theory are introduced and made accessible so that the student can savor the beauty of the primary applications. The later chapters include additional, more demanding topics including Baker's theorem and its applications to the transcendence of special values of L-series; applications of Schneider's theorem and Nesterenko's theorem as they apply to special values of modular forms; and the emerging theory of multiple zeta values. This book is ideal for undergraduates and graduate students, as well as non-expert researchers wishing to learn more about transcendental numbers. The text includes many helpful exercises intended to facilitate practical mastery.

This volume and its companion present a selection of the mathematical writings of P. R. Halmos. The present volume consists of research publications plus two papers which, although of a more expository nature, were deemed primarily of interest to the specialist ("Ten Problems in Hilbert Space" (1970d), and" Ten Years in Hilbert Space" (1979b)). The remaining expository and all the popular writings are in the second volume. The papers in the present volume are arranged chronologically. As it happens, that arrangement also groups the papers according to subject matter: those published before 1950 deal with probability and measure theory, those after 1950 with operator theory. A series of papers from the mid 1950's on algebraic logic is excluded; the papers were republished by Chelsea (New York) in 1962 under the title" Algebraic Logic." This volume contains two introductory essays, one by Nathaniel Friedman on Halmos's work in ergodic theory, one by Donald Sarason on Halmos's work in operator theory. There is an essay by Leonard Gillman on Halmos's expository and popular writings in the second volume. The editors wish to express their thanks to the staff of Springer-Verlag. They are grateful also for the help of the following people: C. Apostol, W. B. Arveson, R. G. Douglas, C. Pearcy, S. Popa, P. Rosenthal, A. L. Shields, D. Voiculescu, S. Walsh. Berkeley, CA Donald E. Sarason vii WORK IN OPERATOR THEORY P. R. Halmos's first papers on Hilbert space operators appeared in 1950.

This book serves as an elementary, self contained introduction into some important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The presentation is made using the classical method of continuation of local solutions with the help of a priori estimates obtained for small data.

Mathematics majors at Michigan State University take a "Capstone" course near the end of their undergraduate careers. The content of this course varies with each offering. Its purpose is to bring together different topics from the undergraduate curriculum and introduce students to a developing area in mathematics. This text was originally written for a Capstone course. Basic wavelet theory is a natural topic for such a course. By name, wavelets date back only to the 1980s. On the boundary between mathematics and engineering, wavelet theory shows students that mathematics research is still thriving, with important applications in areas such as image compression and the numerical solution of differential equations. The author believes that the essentials of wavelet theory are sufficiently elementary to be taught successfully to advanced undergraduates. This text is intended for undergraduates, so only a basic background in linear algebra and analysis is assumed. We do not require familiarity with complex numbers and the roots of unity.

Christian Heinemann explores a unifying model which couples phase separation and damage processes in a system of partial differential equations. The model has technological applications to solder materials where interactions of both phenomena have been observed and cannot be neglected for a realistic description. The author derives the equations in a thermodynamically consistent framework and presents suitable weak formulations for various types of this coupled system. In the main part, he proves the existence of weak solutions and investigates degenerate limits.

Dynamical system theory has developed rapidly over the past fifty years. It is a subject upon which the theory of limit cycles has a significant impact for both theoretical advances and practical solutions to problems. Hopf bifurcation from a center or a focus is integral to the theory of bifurcation of limit cycles, for which normal form theory is a central tool. Although Hopf bifurcation has been studied for more than half a century, and normal form theory for over 100 years, efficient computation in this area is still a challenge with implications for Hilbert's 16th problem.

This book introduces the most recent developments in this field and provides major advances in fundamental theory of limit cycles. Split into two parts, the first focuses on the study of limit cycles bifurcating from Hopf singularity using normal form theory with later application to Hilbert's 16th problem, while the second considers near Hamiltonian systems using Melnikov function as the main mathematical tool.

Classic topics with new results are presented in a clear and concise manner and are accompanied by the liberal use of illustrations throughout. Containing a wealth of examples and structured algorithms that are treated in detail, a good balance between theoretical and applied topics is demonstrated. By including complete Maple programs within the text, this book also enables the reader to reconstruct the majority of formulas provided, facilitating the use of concrete models for study.

Through the adoption of an elementary and practical approach, this book will be of use to graduate mathematics students wishing to study the theory of limit cycles as well as scientists, across a number of disciplines, with an interest in the applications of periodic behavior."